If someone "drives" you crazy, it means the behavior of that person has such a bad effect on you that it makes you feel crazy.
"Makes me crazy" is a more general statement that sometimes has less to do with the person's behavior. As an example, suppose that you are caring for a sick member of your family. Perhaps that person is dying. Perhaps that person is experiencing a lot of pain. It creates a situation that is difficult for you. You might say it is "making you crazy" but you would not say he is "driving you crazy" because it is not his fault.